Arbeitsgruppe Algebraische Geometrie

123-nodale Oktik

The equation of this nice octic (surface of degree 8) is rather simple.
The polynomial \( P (x, y) = (x^2-1)(y^2-1) ((x+y)^2-2)((x-y)^2-2) \)
defines a regular octagon in the plane; the equation of the surface is
\( P (x, y) = 2 (T_8+1) \), where \( T_8 = 128z^8 - 256 x^6 + 160 x^4 -
32 x^{2+1} \) is the Chebyshev-polynomial of degree eight in the
variable \( z \). The surface has in addition 4 singularities of type
\(A_7 \) at infinity. The number of nodes is not particularly high:
Stephan Endrass has constructed an octic with 168 nodes, for which
considerable more ingenuity is needed.

Abelsche Flächen in P^3

Abelian varieties do not fit easily in projective
space. A line bundle of type (1,4) on an abelian surface defines a
mapping of the surface to \( P^3 \) and gives a birational image as a
hypersurface of degree 8. The pictures show the real locus of the octic,
which consist of two real immersed tori, connected by naked lines of
singularities. Note that one clearly sees the torus, its three
parameters and the polarisation type! For more details see Abelian surfaces of type (1,4)